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%% lecture11.tex
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%% Made by alex
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%% Started on  Thu Jan  5 08:13:35 2012 alex
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\exercises
\begin{xca}
Let us consider chain complex $C=C_k$ where $C_k=0$ for
$k\not=n,n-1$, $C_n=C_{n-1}=\ZZ$ and the boundary operator
$C_{n}\to C_{n-1}$ acts as multiplication by $m$. Calculate
corresponding homology and cohomology with coefficients in the
group $G$. In particular, consider the cases when $G$ is an
infinite or finite cyclic group.
\end{xca}
\begin{xca}
Calculate cohomology groups with integer coefficients and with
coefficients in cyclic group $\ZZ_k$ for the following spaces.
\begin{enumerate}
\item The quotient of $S^2$ obtained by identifying north and
  south poles to a point.
\item The quotient of $S^2$ under identification of $x$ with $-x$
  for $x$ in the equator $S^1$.
\end{enumerate}
\end{xca}
\begin{xca}
Calculate relative cohomology $H_k (\RP^5, \RP^2; \ZZ)$ using
cell decomposition of $\RP^5$. (The projective space $\RP^2$ is
embedded into $\RP^5$ in standard way.) Describe the exact
cohomology sequence of the pair $(\RP^5, \RP^2)$ and check that
it is exact.
\end{xca}

